1. **Problem Statement:** We are given a trapezoidal plate with the top side length $33$ inches, the bottom side length $10$ inches, and the left side length $24$ inches. We need to find the angle between the left side and the top side. 2. **Understanding the Problem:** The trapezoid has two parallel sides (top and bottom). The left side is slanting, forming an angle with the top side. We want to find this angle. 3. **Approach:** We can model the trapezoid in a coordinate system: - Place the top side along the x-axis from $(0,0)$ to $(33,0)$. - The bottom side is parallel to the top side and shorter, length $10$. - The left side connects the top left corner $(0,0)$ to the bottom left corner, which we need to find. 4. **Finding the horizontal distance of the left side:** Since the bottom side is $10$ inches and the top side is $33$ inches, the bottom side is shifted to the right by some horizontal distance $x$. 5. **Using the Pythagorean theorem:** The left side length is $24$ inches, which is the hypotenuse of a right triangle with vertical leg $24$ inches and horizontal leg $x$ (the horizontal shift). 6. **Calculate the horizontal shift $x$:** The vertical leg is the height $h$ of the trapezoid, which we don't know yet. But since the bottom side is shorter, the difference in length between top and bottom is $33 - 10 = 23$ inches. 7. **Assuming the bottom side is shifted right by $x$, the horizontal leg of the left side is $x$, and vertical leg is $h$. So:** $$24^2 = x^2 + h^2$$ 8. **The bottom side is parallel to the top side, so the height $h$ is the vertical distance between the two parallel sides. The bottom side starts at $(x, h)$ and ends at $(x+10, h)$. The top side is at $y=0$ from $(0,0)$ to $(33,0)$. 9. **Since the bottom side is shifted right by $x$, the difference in horizontal length between top and bottom is $23$, so:** $$x + 10 = 33 \implies x = 23$$ 10. **Now substitute $x=23$ into the Pythagorean theorem:** $$24^2 = 23^2 + h^2$$ $$576 = 529 + h^2$$ $$h^2 = 576 - 529 = 47$$ $$h = \sqrt{47}$$ 11. **Calculate the angle $\theta$ between the left side and the top side:** The angle is the angle between the hypotenuse (left side) and the horizontal leg (top side), so: $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{24} = \frac{23}{24}$$ 12. **Calculate $\theta$:** $$\theta = \arccos\left(\frac{23}{24}\right)$$ 13. **Final answer:** $$\theta \approx \arccos(0.9583) \approx 16.26^\circ$$ **Therefore, the angle between the left side and the top side is approximately $16.26^\circ$.**